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Surface Area of a Hemisphere

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A hemisphere is a 3D shape that is half of a sphere’s volume and surface area. The surface area of a hemisphere comprises both the curved region and the base area combined.

  • Hemisphere’s Total Surface Area (TSA) = Curved Surface Area + Base Area = 3Ï€r² square units.
  • Curved Surface Area (CSA) = 2Ï€r² square units,
    where, r is the radius of the hemisphere.

It has a flat bottom surface and a rounded top surface, like a bowl or the inside of a hollow ball that has been cut in half. You can get a hemisphere by slicing a sphere horizontally or vertically through its centre, resulting in two identical halves with the same diameter but different heights. One direction (the slice) will have half the diameter, while the other direction will match the full diameter of the original sphere.

In this article, we will learn in detail about the calculation of volume and surface areas of a hemisphere, its derivation, and some solved examples.

What is a Hemisphere?

 A hemisphere is formed when a plane divides a sphere into two equal parts. In other words, a hemisphere is exactly half of a sphere in geometry. It is made up of two parts: “hemi,” meaning half, and “sphere,” which is a three-dimensional mathematical shape for round objects. When a sphere is sliced at the exact centre along its diameter, two equal hemispheres are generated. Therefore, a hemisphere is a three-dimensional geometric object consisting of half of a sphere with one side flat and the other side shaped like a round bowl.

Sphere-and-hemisphere

Sphere and Hemisphere

Real-Life Examples of Hemispheres

The examples of hemispheres can be seen in everyday life. For instance, the bowl that we use to eat food from is nothing but a hollow hemisphere. The half-cut shell of the coconut is an example of a hollow hemisphere. When we cut round-shaped fruits like oranges, tamarind, watermelon, etc., the shape of the fruit becomes a solid hemispherical shape. Below shown images are real-life examples of hemispheres.

real-life examples of hemisphere

Real-life Examples of Hemisphere

What is Surface Area of a Hemisphere?

The surface area of a hemisphere is the total surface area of a hemisphere. The Hemispherical area is defined by two types of hemispheres, the solid hemisphere, and the hollow hemisphere. The surface area can be found in two ways:

  • Curved Surface Area of a Hemisphere (CSA).
  • Total Surface Area of a Hemisphere (TSA).

Curved Surface Area of Hemisphere Formula

The curved surface area of the hemisphere formula is defined as the area covered by its curved surface. It is equal to half of the total surface area of a sphere. The formula for the curved surface area of the hemisphere equals two times the product of pi and the square of the radius of the hemisphere.

Curved Surface Area of a Hemisphere = 2Ï€r2

where,

  • Ï€ is a constant with the value of 3.14, and
  • r is the radius of hemisphere.

Derivation of the Formula of Curved Surface Area of a Hemisphere

Curved Surface Area for Hemisphere is the area of all the curved surfaces of the hemisphere which is only half of the spherically curved surface as the base of the hemisphere is a flat surface that is not curved. Thus,

Curved Surface Area of a Hemisphere = 1/2 × (Curved Surface Area of Sphere)

CSA = 1/2 (4Ï€r2)

CSA = 2Ï€r2

Base Area of Hemisphere

The base of the hemisphere is in a circular shape, and therefore, the formula for the base area of the hemisphere is equal to the area of a circle.

Base Area of Hemisphere = πr2

Total Surface Area of a Hemisphere Formula

The total surface area of a hemisphere is defined as the total space covered by the surface of the hemisphere. The total surface area is given by the sum of its curved surface area and base area. The formula for total surface area equals three times the product of the pi (Ï€) and the square of the radius of the hemisphere.

Total surface area of a hemisphere

Total Surface Area of Hemisphere

Total Surface Area of Hemisphere = 3Ï€r2

where,

  • Ï€ is a constant with the value of 3.14, and
  • r is the radius of hemisphere.

Derivation of the Formula of Total Surface Area of a Hemisphere

Total surface area for a hemisphere is the sum of the curved surface area of the hemisphere and the area of its circular base since a hemisphere is just half of a sphere with a circular base. Therefore, the total surface area of a hemisphere can be expressed as:

Total Surface Area of a Hemisphere = Curved Surface Area of Hemisphere + Base Area of Hemisphere

⇒ TSA = 2πr2 + πr2

TSA = 3Ï€r2

Surface Area of a Hollow Hemisphere Formula

Surface area of a hollow hemisphere can be understood by considering its components. A hollow hemisphere possesses two diameters, as the presence of the inner hollow hemisphere introduces a smaller diameter. Observing closely, the surface area of a hollow hemisphere comprises three main parts:

  1. Curved surface area of Outer Hemisphere.
  2. Curved surface area of Inner Hemisphere.
  3. Area of Remaining Base.

Derivation of the Formula Surface Area of a Hollow Hemisphere

Let’s break down the areas in order to obtain the surface area of a hollow hemisphere:

  • Crved Surface area of Outer Hemisphere = 2Ï€R2
  • Curved Surface area of Inner Hemisphere = 2Ï€r2
  • Base Area of Hollow Hemisphere = Ï€(R2 – r2)

Therefore, total surface area of hemisphere = 2Ï€R2 + 2Ï€r2 + Ï€(R2 – r2)

TSA = 2Ï€R2 + 2Ï€r2 + Ï€R2 – Ï€r2

TSA = 3πR2 + πr2

Surface Area of a Hollow Hemisphere

Surface Area of a Hollow Hemisphere

Total Surface Area of a Hollow Hemisphere with Closed Base

The total surface area of a hollow hemisphere with a closed base consists of the following components:

  1. Curved Surface Area of Larger Hemisphere: It is calculated using the formula for the surface area of a sphere: 2Ï€R 2, where R is the radius of the larger hemisphere.
  2. Curved Surface Area of the Smaller Hemisphere: Similarly, the curved surface area of the smaller hemisphere is also 2Ï€r2, where r is the radius of the smaller hemisphere.
  3. Area of Remaining Base: Base area of the hollow hemisphere is the difference between the areas of the bases of the larger and smaller hemispheres. The area of the larger base is 2Ï€R 2, and the area of the smaller base is 2Ï€r2.

Total surface area of the hollow hemisphere is the sum of these three components:

Total Surface Area = 2πR 2 + 2πr2 + (πR 2 − πr2)

= 2πR 2 + 2πr2 + πR 2 − πr2

= 3πR 2 + πr2

This formula accounts for the surface area of both the curved portions and the closed base of the hollow hemisphere.

How to Find Surface Area of a Hemisphere?

The surface area of a hemisphere can be found by following easy steps based on what type of hemisphere is given. If a solid hemisphere is given, the formula of total surface area and curved surface area can be used based on the requirement, and if a hollow hemisphere is given, the formula for a hollow hemisphere must be used. Following are the steps that can be followed to obtain the surface areas based on the requirement.

How to Find Curved Surface Area of a Hemisphere

The formula for the curved surface area of a hemisphere when the given radius is “r” is 2Ï€r2. Below are the steps provided to find the curved surface area of a hemisphere:

  • Note down the radius of the hemisphere.
  • Put the “r” value in the formula for the curved surface area of a sphere, that is, CSA = 2Ï€r2.
  • Present the final answer in square units.

Example: Calculate the curved surface area of a hemisphere radius of 5 m. (Use π = 3.14).

Solution:

We have,

r = 5

Using the formula we get,

CSA = 2Ï€r2

⇒ CSA = 3 (3.14) (5)2

⇒ CSA = 235.5 sq. m

How to Find the Total Surface Area of a Hemisphere

The formula for the total surface area of a hemisphere when the given radius is “r” is 3Ï€r2. Below are the steps provided to find the curved surface area of a hemisphere:

  • Note down the radius of the hemisphere.
  • Put the “r” value in the formula for the total surface area of a sphere, that is, TSA = 3Ï€r2.
  • Present the final answer in square units.

Example: Calculate the total surface area of a hemisphere diameter of 16cm.

Solution:

We have,

  • d = 16cm
  • r = 8cm

Using the formula we get,

TSA = 3Ï€r2

⇒ TSA = 3 (3.14) (8)2

⇒ TSA = 602.88 sq. cm.

How to Find Surface Area of a Hollow Hemisphere

The formula for the surface area of a hollow hemisphere when the given radius is “r” is 3Ï€R2 + Ï€r2. Below are the steps provided to find the surface area of a hollow hemisphere:

  • Note down the radius of the hemisphere.
  • Put the “r” value in the formula for the total surface area of a sphere, that is, TSA = 3Ï€R2 + Ï€r2.
  • Present the final answer in square units.

Summary of Surface Area of Hemisphere

A hemisphere is also called a semi-sphere or half a sphere. It is formed when a plane divides a sphere into two equal parts. When a sphere is sliced at the exact centre along its diameter, two equal hemispheres are generated. The surface area of a hemisphere, also known as the total surface area of a hemisphere, is the sum of all the areas of its faces, including the curved surface and the base. 

  • CSA of Hemisphere = 2Ï€r2

The base area of a hemisphere is the area of the flat surface at the bottom of the hemisphere.

  • Base area of Hemisphere = Ï€r2
  • Total Surface Area (TSA) of Hemisphere = 3Ï€r2

The surface area of a hollow hemisphere can be calculated using the following formula the sum of the inner surface, an outer surface, and the area of the base which is in the shape of a ring.

  • Curved Surface Area of a Hollow Hemisphere =  2Ï€(R2 + r2)
  • Total Surface Area of a Hollow Hemisphere = 3Ï€R2 + Ï€r2

Where r is the radius of the hemisphere and in the case of the hollow hemisphere R and r are the outer and inner radius of the hollow hemisphere.

Solved Questions on Surface Area of Hemisphere

Question 1: Calculate the total surface area of a hemisphere radius of 4 m.

Solution:

We have,

r = 4

Using the formula we get,

TSA = 3Ï€r2

⇒ TSA = 3 (3.14) (4)2

⇒ TSA = 150.72 sq. m

Question 2: Calculate the radius of a hemisphere if its total surface area is 200 sq. m.

Solution:

We have,

A = 200

Using the formula we get,

A = 3Ï€r2

⇒  r2 = A/3π

⇒  r2 = 200/3 (3.14)

⇒  r = 4.60 m

Question 3: Calculate the radius of a hemisphere if its total surface area is 350 sq. m.

Solution:

We have,

A = 200

Using the formula we get,

A = 3Ï€r2

⇒  r2 = A/3π

⇒  r2 = 350/3 (3.14)

⇒  r = 6.09 m

Question 4: Calculate the curved surface area of a hemisphere radius of 4 m.

Solution:

We have,

r = 4

Using the formula we get,

CSA = 2Ï€r2

⇒ CSA = 3 (3.14) (4)2

⇒ CSA = 150.72 sq. m

Question 5: Calculate the radius of a hemisphere if its curved surface area is 790 sq. m.

Solution:

We have,

A = 790

Using the formula we get,

A = 2Ï€r2

⇒ r2 = A/2π

⇒ r2 = 350/2 (3.14)

⇒ r = 7.46 m

Hemisphere Surface Area Practice Questions

High Order Thinking Skills Questions are very important to hone your understanding and conceptual clarity about the topic. Solve the following HOTS questions on Surface Area of a Hemisphere to build conceptual confidence:

Question 1: A dome-shaped building has a radius of 10 meters. The dome is painted on both the inside and the outside. What is the total area that gets painted? How would the paint required change if the radius of the dome was increased by 10%?

Question 2: A company manufactures hemispherical bowls with a radius of 5 cm. If the cost of the material used to make the bowl is proportional to its surface area, how would the cost change if the company decided to manufacture bowls that are 20% larger in radius?

Question 3: A planet is approximately a sphere. If we consider the Northern Hemisphere, what would be the change in its surface area if the radius of the planet increased by 1%?

Question 4: A hemispherical tank with a radius of 2 meters is used to store water. If the tank is expanded by increasing its radius by 50%, how much more water can it store? How does this relate to the change in the surface area of the tank?

Question 5: An igloo is built in the shape of a hemisphere. If the radius of the igloo is doubled, how does this affect the surface area of the igloo? How would this change affect the amount of ice needed to build the igloo?

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FAQs on Surface Area of a Hemisphere

What is a hemisphere?

A hemisphere is a three-dimensional shape that is half of a sphere. It has a curved surface and a flat base.

What units are used to measure the surface area of a hemisphere?

The surface area of a hemisphere can be measured in square units, such as square centimeters (cm²) or square meters (m²).

What is the curved surface area of a hemisphere?

The curved surface area of hemisphere is another name for the lateral surface area of hemisphere and the formula for curved surface area of a hemisphere is 2Ï€r2.

What is the area of the base of a hemisphere?

The base area of hemisphere has circular shape and therefore, the area of circle will be used to find the base area of hemisphere. Hence, base area becomes πr2.

What is the formula for the total surface area of a hemisphere?

The formula for the total surface area of a hemisphere is 3Ï€r2, where r is the radius of the hemisphere.

How to find the surface area of a hemisphere?

The formula for total surface area of a hemisphere when the given radius is “r” is 3Ï€r2. Below are the steps provided to find curved surface area of a hemisphere:

  • Note down the radius of the hemisphere.
  • Put the “r” value in the formula for total surface area of sphere, that is, TSA = 3Ï€r2.
  • Present the final answer in square units.

Explain the difference between the curved surface area of a hemisphere and the surface area of a hemisphere.

The curved surface area of hemisphere is only the area covered by the curved surface and the formula for curved surface area of hemisphere is 2Ï€r2. While the surface area of a hemisphere is the sum of the curved surface area and the base area of hemisphere, therefore, the formula for TSA becomes 3Ï€r2.

What is the surface area of a hemispherical shell?

Yes, the formula for the surface area of a hemisphere can be used for any size of hemisphere, as long as the radius is known.

How does the surface area of a hemisphere compare to the surface area of a sphere?

The surface area of a hemisphere is half of the surface area of a full sphere with the same radius.



Last Updated : 21 Feb, 2024
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